Theorem isoml 37179

Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isoml.b	⊢ 𝐵 = (Base‘𝐾)
isoml.l	⊢ ≤ = (le‘𝐾)
isoml.j	⊢ ∨ = (join‘𝐾)
isoml.m	⊢ ∧ = (meet‘𝐾)
isoml.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
isoml	⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦

Allowed substitution hints:   ∨ (𝑥,𝑦)   ≤ (𝑥,𝑦)   ∧ (𝑥,𝑦)   ⊥ (𝑥,𝑦)

Proof of Theorem isoml

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6756	. . . 4 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isoml.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2797	. . 3 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		isoml.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2797	. . . . . 6 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5081	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
8		fveq2 6756	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		isoml.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2797	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11		eqidd 2739	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
12		fveq2 6756	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13		isoml.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
14	12, 13	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
15		eqidd 2739	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑦 = 𝑦)
16		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17		isoml.o	. . . . . . . . . 10 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
19	18	fveq1d 6758	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( ⊥ ‘𝑥))
20	14, 15, 19	oveq123d 7276	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑥)))
21	10, 11, 20	oveq123d 7276	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))
22	21	eqeq2d 2749	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
23	7, 22	imbi12d 344	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
24	3, 23	raleqbidv 3327	. . 3 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
25	3, 24	raleqbidv 3327	. 2 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
26		df-oml 37120	. 2 ⊢ OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
27	25, 26	elrab2 3620	1 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  occoc 16896  joincjn 17944  meetcmee 17945  OLcol 37115  OMLcoml 37116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-oml 37120

This theorem is referenced by:  isomliN  37180  omlol  37181  omllaw  37184